I tend to believe that plugging my puzzly, escape-room-y blog would not generally be wildly welcome here; that said, my most recent post, on the titular question, may be of interest.

Executive summary: aside from the question of whether or not you'd want to, while the answer to the first half of the question is that you can't, the answer to the second half ("...or something like it?") seems to me to have more to it than you might think.

I've thought about this a bit, and it's a tricky one. Whilst the olympics may well have a wonderful ideal to it, I think it's also fair to say that Faster, Higher, Stronger has proven to have the ability of capturing the imagination of the wider public who choose to be entertained by the games.

The catch 22 seems to be how to capture public imagination. Seemingly you can't capture it unless you are perceived as a big deal, but how can you possibly be a big deal without public imagination? I don't think any kind of "mindsport" comes close to this - perhaps video games in places like Korea comes closest.

I think part of the issue for us is with terminology - using any word vaguely relating to sport will draw the response "that's not a real sport". I'm not sure how else to describe generalised competitive activity (this definitely includes things like video games) though. Another part of the equation is money. Ultimately this has to come from somewhere if you want something to take off.

That all said, I think there is something to be said for remaining "ideologically sound" and not compromising or selling out. I don't think very many people at all get the idea of competitive puzzle solving, but I don't think this should take anything away from the impressiveness of those who are good at it. Maybe some kind of compromise is necessary, perhaps dependent on who is stumping up the cash, but in the absence of a clear plan/strategy/vision I think such a compromise would be a pity.

In sport, the money brought into the game is roughly proportional to the spectator interest. (both in person, and via media views)
This is a self-fuelling cycle, as more money means a better show can be put on.

I've seen a few things become popular on this basis, often with Red Bull associated with them - eg, soapbox racing, cliff diving, monster trucks, (stunt) air races, robot wars.

Other sports have huge popularity, but a very low spectator level. These will get some coverage, but not mega-money. eg, fishing, chess, go, scrabble, crosswords.

Without sounding pessimistic, I think the best sudoku/puzzles could ever do is to fall into the latter category, but even that is going to take some doing so that the cream of the crop can make a living off 'playing' it.